satisfying given conditions. 



167 



Let the direction cosines of the axes of the curve be given by the 

 scheme 



Tangent l 1? m 1} n 1} 



Prin. Norm. L, m 2 , n if 



Binormal l 3 , m 3 , n 3 . 



Then for the indicatrix 



X = h COS \jr + l 3 sin yjr, 

 therefore dx — L [de cos -v/r — dr sin yfr) 



by the Serret-Frenet formulae ; 

 thus ds — de cos yjr — dr sin -v/r. 



Now x ds' — [l L cos ty + l 3 sin -v/r] [de cos yjr — dr sin yfr] , 

 and 

 z dy — y dz = [(niin, — « 2 '?^i) cos ty + (n 3 m, — n. 2 m 3 ) sin i/r] cfo 

 = [— ? 3 cos ty + li sin -v/r] [cZe cos yjr — dr sin i/r] ; 

 therefore 



x ds cos i|r + (z dy — y dz (l ) sin ty = li [de cos i/r — c£t sin -v/r] . 



cos -v/r sin °fr\ 

 P 



~ . , cos 4r sin •vjr] I a 



Consider u = y -[ F 



[ p a ) cos 



t|t sin -v/r 

 /a a 



this implies that for the new curve 



( cos ty sin yjr) „ 

 I P <r ) 



COS l|r sin l/r 



r = l. 



COS \|r sin -v/r 

 V P <r ) 



The equations of this curve, viz. 



x = JLuds, 

 at once reduce to the form 



x ' = jl\ [cos yfrde — sin y\r dr} F (tan ^), 

 or ®=f [^o^^o cos i|r + (z dy — y Q dz ) sin -^r] i* 1 (tan ^). 



The object of introducing the angle yjr is to make the function F 

 as simple as possible. 



When ^ = 0, 



x = Jx ds F (tan x) 



